An understanding of our fundamental limitations is among the most important contributions of science and of mathematics. At the same time, various fundamental limitations stated by many great minds turned out to be wrong, sometimes rather quickly.

There are quite a few cases where things that were considered to be impossible turned out to be possible. Immanuel Kant claimed: “No finite Reason can hope to understand the production of even a blade of grass by mere mechanical causes.” This quote is from the Critique of Judgment (1790). Elsewhere Kant wrote: “It is absurd to hope that another Newton will arise in the future who shall make comprehensible by us the production of a blade of grass according to natural laws which no design has ordered.”

Auguste Comte claimed: “Of all objects, the planets are those which appear to us under the least varied aspect. We see how we may determine their forms, their distances, their bulk, and their motions, but we can never know anything of their chemical or mineralogical structure; and, much less, that of organized beings living on their surface …” (The Positive Philosophy, Book II, Chapter 1 (1842)).

Spectroscopy was developed by Gustav Kirchhoff in the 1840s, and the first spectroscopic analysis of the sun appeared about ten years later, less than 20 years after Comte’s statement.

A slightly different example relates to the philosopher Wittgenstein. As usual he is more cryptic. He gives an example of a person making claims that seem crazy to us. Writing before expeditions to the moon became a reality, Wittgenstein speaks of a person who claims that “men sometimes visit the moon.” The following excerpt is taken from “On Certainty.”

“What we believe depends upon what we learn. We all believe that it isn’t possible to get to the moon; but there might be people who believe that it is possible and that it sometimes happens. We say: these people do not know a lot that we know. And, let them be never so sure of their belief, they are wrong and we know it.”

There are famous quotations of predictions made by Lord Kelvin. Kelvin did not believe that heavier-than-air flying machines were possible and he regarded X-rays as a hoax. Kelvin’s ingenuity was manifested even in cases where his overall predictions were wrong. He gave a lecture on the state of physics at the turn of the twentieth century, and – not unlike Hilbert’s famous lectures in mathematics – claimed that physics was nearly complete and all problems would soon be settled. He mentioned, however, “two clouds on the horizon,” the unexpected behavior of ether in the Michelson-Morley experiment and the problem of the spectrum of black body radiation. His genius as a physicist was manifested by the fact that of all the scores of open problems in physics present at the time (as there always are), he pinpointed the two problems that subsequently led to revolutions: the ether problem led to relativity, and black body radiation to quantum theory.

Itamar Pitowsky

(The items and citations of Kant, Comte, and Kelvin were contributed by Itamar Pitowsky, a philosophy professor at the Hebrew University of Jerusalem and my former classmate there. Itamar also drew my attention, in the summer of 2006, to the weblog discussions regarding string theory.)